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Elementary Algebra 2e

1.5 Visualize Fractions

Elementary Algebra 2e1.5 Visualize Fractions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Learning Objectives

By the end of this section, you will be able to:
  • Find equivalent fractions
  • Simplify fractions
  • Multiply fractions
  • Divide fractions
  • Simplify expressions written with a fraction bar
  • Translate phrases to expressions with fractions
Be Prepared 1.5

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions.

Find Equivalent Fractions

Fractions are a way to represent parts of a whole. The fraction 1313 means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See Figure 1.11. The fraction 2323 represents two of three equal parts. In the fraction 23,23, the 2 is called the numerator and the 3 is called the denominator.

Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.
Figure 1.11 The circle on the left has been divided into 3 equal parts. Each part is 1313 of the 3 equal parts. In the circle on the right, 2323 of the circle is shaded (2 of the 3 equal parts).

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Model Fractions” will help you develop a better understanding of fractions, their numerators and denominators.

Fraction

A fraction is written ab,ab, where b0b0 and

  • a is the numerator and b is the denominator.

A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included.

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate 6666 pieces, or, in other words, one whole pie.

A circle is shown and is divided into six section. All sections are shaded.

So 66=1.66=1. This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.

Property of One

aa=1(a0)aa=1(a0)

Any number, except zero, divided by itself is one.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Fractions Equivalent to One” will help you develop a better understanding of fractions that are equivalent to one.

If a pie was cut in 66 pieces and we ate all 6, we ate 6666 pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate 8888 pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions 6666 and 8888 have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. Figure 1.12 shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that 1212 is equivalent to 48.48. In other words, they are equivalent fractions.

A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.
Figure 1.12 Since the same amount is of each pizza is shaded, we see that 1212 is equivalent to 48.48. They are equivalent fractions.

Equivalent Fractions

Equivalent fractions are fractions that have the same value.

How can we use mathematics to change 1212 into 48?48? How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into 88 pieces instead of just 2. Mathematically, what we’ve described could be written like this as 1·42·4=48.1·42·4=48. See Figure 1.13.

A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.
Figure 1.13 Cutting each half of the pizza into 44 pieces, gives us pizza cut into 8 pieces: 1·42·4=48.1·42·4=48.

This model leads to the following property:

Equivalent Fractions Property

If a,b,ca,b,c are numbers where b0,c0,b0,c0, then

ab=a·cb·cab=a·cb·c

If we had cut the pizza differently, we could get

An image shows three rows of fractions. In the first row are the fractions “1, times 2, divided by 2, times 2, equals two fourths”. Next to this is the word “so” and the fraction “one half, equals two fourths. The second row reads “1, times 3, divided by 2 times 3, equals three sixths”. Next to this is the word “so” and the fraction “one half equals, three sixths”. The third row reads “1 times 10, divided by 2 times 10, ten twentieths”. Next to this is the word “so” and the fraction “one half equals, ten twentieths”.

So, we say 12,24,36,and102012,24,36,and1020 are equivalent fractions.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Equivalent Fractions” will help you develop a better understanding of what it means when two fractions are equivalent.

Example 1.64

Find three fractions equivalent to 25.25.

Try It 1.127

Find three fractions equivalent to 35.35.

Try It 1.128

Find three fractions equivalent to 45.45.

Simplify Fractions

A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.

For example,

  • 2323 is simplified because there are no common factors of 2 and 3.
  • 10151015 is not simplified because 55 is a common factor of 10 and 15.

Simplified Fraction

A fraction is considered simplified if there are no common factors in its numerator and denominator.

The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.

In Example 1.64, we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.

Equivalent Fractions Property

If a,b,ca,b,c are numbers where b0,c0,b0,c0,

thenab=a·cb·canda·cb·c=abthenab=a·cb·canda·cb·c=ab

Example 1.65

Simplify: 3256.3256.

Try It 1.129

Simplify: 4254.4254.

Try It 1.130

Simplify: 4581.4581.

Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.

Example 1.66 How to Simplify a Fraction

Simplify: 210385.210385.

Try It 1.131

Simplify: 69120.69120.

Try It 1.132

Simplify: 120192.120192.

We now summarize the steps you should follow to simplify fractions.

How To

Simplify a Fraction.

  1. Step 1. Rewrite the numerator and denominator to show the common factors.
    If needed, factor the numerator and denominator into prime numbers first.
  2. Step 2. Simplify using the equivalent fractions property by dividing out common factors.
  3. Step 3. Multiply any remaining factors, if needed.

Example 1.67

Simplify: 5x5y.5x5y.

Try It 1.133

Simplify: 7x7y.7x7y.

Try It 1.134

Simplify: 3a3b.3a3b.

Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Model Fraction Multiplication” will help you develop a better understanding of multiplying fractions.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with 34.34.

A rectangle made up of four squares in a row. The first three squares are shaded.

Now we’ll take 1212 of 34.34.

A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.

Notice that now, the whole is divided into 8 equal parts. So 12·34=38.12·34=38.

To multiply fractions, we multiply the numerators and multiply the denominators.

Fraction Multiplication

If a,b,candda,b,candd are numbers where b0andd0,b0andd0, then

ab·cd=acbdab·cd=acbd

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In Example 1.68, we will multiply negative and a positive, so the product will be negative.

Example 1.68

Multiply: 1112·57.1112·57.

Try It 1.135

Multiply: 1028·815.1028·815.

Try It 1.136

Multiply: 920·512.920·512.

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as a1.a1. So, for example, 3=31.3=31.

Example 1.69

Multiply: 125(−20x).125(−20x).

Try It 1.137

Multiply: 113(−9a).113(−9a).

Try It 1.138

Multiply: 137(−14b).137(−14b).

Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of 2323 is 32.32.

Notice that 23·32=1.23·32=1. A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of 107107 is 710,710, since 107(710)=1.107(710)=1.

Reciprocal

The reciprocal of abab is ba.ba.

A number and its reciprocal multiply to one ab·ba=1.ab·ba=1.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Model Fraction Division” will help you develop a better understanding of dividing fractions.

To divide fractions, we multiply the first fraction by the reciprocal of the second.

Fraction Division

If a,b,candda,b,candd are numbers where b0,c0andd0,b0,c0andd0, then

ab÷cd=ab·dcab÷cd=ab·dc

To divide fractions, we multiply the first fraction by the reciprocal of the second.

We need to say b0,c0andd0b0,c0andd0 to be sure we don’t divide by zero!

Example 1.70

Divide: 23÷n5.23÷n5.

Try It 1.139

Divide: 35÷p7.35÷p7.

Try It 1.140

Divide: 58÷q3.58÷q3.

Example 1.71

Find the quotient: 78÷(1427).78÷(1427).

Try It 1.141

Find the quotient: 727÷(3536).727÷(3536).

Try It 1.142

Find the quotient: 514÷(1528).514÷(1528).

There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.

  • “To multiply fractions, multiply the numerators and multiply the denominators.”
  • “To divide fractions, multiply the first fraction by the reciprocal of the second.”

Another way is to keep two examples in mind:

This is an image with two columns. The first column reads “One fourth of two pizzas is one half of a pizza. Below this are two pizzas side-by-side with a line down the center of each one representing one half. The halves are labeled “one half”. Under this is the equation “2 times 1 fourth”. Under this is another equation “two over 1 times 1 fourth.” Under this is the fraction two fourths and under this is the fraction one half. The next column reads “there are eight quarters in two dollars.” Under this are eight quarters in two rows of four. Under this is the fraction equation 2 divided by one fourth. Under this is the equation “two over one divided by one fourth.” Under this is two over one times four over one. Under this is the answer “8”.

The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.

Complex Fraction

A complex fraction is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

6733458x2566733458x256

To simplify a complex fraction, we remember that the fraction bar means division. For example, the complex fraction 34583458 means 34÷58.34÷58.

Example 1.72

Simplify: 3458.3458.

Try It 1.143

Simplify: 2356.2356.

Try It 1.144

Simplify: 37611.37611.

Example 1.73

Simplify: x2xy6.x2xy6.

Try It 1.145

Simplify: a8ab6.a8ab6.

Try It 1.146

Simplify: p2pq8.p2pq8.

Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

To simplify the expression 537+1,537+1, we first simplify the numerator and the denominator separately. Then we divide.

537+1537+1
2828
1414

How To

Simplify an Expression with a Fraction Bar.

  1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
  2. Step 2. Simplify the fraction.

Example 1.74

Simplify: 42(3)22+2.42(3)22+2.

Try It 1.147

Simplify: 63(5)32+3.63(5)32+3.

Try It 1.148

Simplify: 44(6)32+3.44(6)32+3.

Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.

−13=13negativepositive=negative1−3=13positivenegative=negative−13=13negativepositive=negative1−3=13positivenegative=negative

Placement of Negative Sign in a Fraction

For any positive numbers a and b,

ab=ab=abab=ab=ab

Example 1.75

Simplify: 4(−3)+6(−2)−3(2)2.4(−3)+6(−2)−3(2)2.

Try It 1.149

Simplify: 8(−2)+4(−3)−5(2)+3.8(−2)+4(−3)−5(2)+3.

Try It 1.150

Simplify: 7(−1)+9(−3)−5(3)2.7(−1)+9(−3)−5(3)2.

Translate Phrases to Expressions with Fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of aa and bb is the result we get from dividing aa by b,b, or ab.ab.

Example 1.76

Translate the English phrase into an algebraic expression: the quotient of the difference of m and n, and p.

Try It 1.151

Translate the English phrase into an algebraic expression: the quotient of the difference of a and b, and cd.

Try It 1.152

Translate the English phrase into an algebraic expression: the quotient of the sum of pp and q,q, and rr

Section 1.5 Exercises

Practice Makes Perfect

Find Equivalent Fractions

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

343.

3838

344.

5858

345.

5959

346.

1818

Simplify Fractions

In the following exercises, simplify.

347.

40884088

348.

63996399

349.

1086310863

350.

1044810448

351.

120252120252

352.

182294182294

353.

3x12y3x12y

354.

4x32y4x32y

355.

14x221y14x221y

356.

24a32b224a32b2

Multiply Fractions

In the following exercises, multiply.

357.

34·91034·910

358.

45·2745·27

359.

23(38)23(38)

360.

34(49)34(49)

361.

59·31059·310

362.

38·41538·415

363.

(1415)(920)(1415)(920)

364.

(910)(2533)(910)(2533)

365.

(6384)(4490)(6384)(4490)

366.

(3360)(4088)(3360)(4088)

367.

4·5114·511

368.

5·835·83

369.

37·21n37·21n

370.

56·30m56·30m

371.

−8(174)−8(174)

372.

(−1)(67)(−1)(67)

Divide Fractions

In the following exercises, divide.

373.

34÷2334÷23

374.

45÷3445÷34

375.

79÷(74)79÷(74)

376.

56÷(56)56÷(56)

377.

34÷x1134÷x11

378.

25÷y925÷y9

379.

518÷(1524)518÷(1524)

380.

718÷(1427)718÷(1427)

381.

8u15÷12v258u15÷12v25

382.

12r25÷18s3512r25÷18s35

383.

−5÷12−5÷12

384.

−3÷14−3÷14

385.

34÷(−12)34÷(−12)

386.

−15÷(53)−15÷(53)

In the following exercises, simplify.

387.

82112358211235

388.

91633409163340

389.

452452

390.

53105310

391.

m3n2m3n2

392.

38y1238y12

Simplify Expressions Written with a Fraction Bar

In the following exercises, simplify.

393.

22+31022+310

394.

19461946

395.

482415482415

396.

464+4464+4

397.

−6+68+4−6+68+4

398.

−6+3178−6+3178

399.

4·36·64·36·6

400.

6·69·26·69·2

401.

4212542125

402.

72+16072+160

403.

8·3+2·914+38·3+2·914+3

404.

9·64·722+39·64·722+3

405.

5·63·44·52·35·63·44·52·3

406.

8·97·65·69·28·97·65·69·2

407.

523235523235

408.

624246624246

409.

7·42(85)9·33·57·42(85)9·33·5

410.

9·73(128)8·76·69·73(128)8·76·6

411.

9(82)3(157)6(71)3(179)9(82)3(157)6(71)3(179)

412.

8(92)4(149)7(83)3(169)8(92)4(149)7(83)3(169)

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

413.

the quotient of r and the sum of s and 10

414.

the quotient of A and the difference of 3 and B

415.

the quotient of the difference of xandy,and3xandy,and3

416.

the quotient of the sum of mandn,and4qmandn,and4q

Everyday Math

417.

Baking. A recipe for chocolate chip cookies calls for 3434 cup brown sugar. Imelda wants to double the recipe. How much brown sugar will Imelda need? Show your calculation. Measuring cups usually come in sets of 14,13,12,and114,13,12,and1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the cookie recipe.

418.

Baking. Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs 2323 cup of condensed milk. How much condensed milk will Nina need? Show your calculation. Measuring cups usually come in sets of 14,13,12,and114,13,12,and1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk needed for 44 pans of fudge.

419.

Portions Don purchased a bulk package of candy that weighs 55 pounds. He wants to sell the candy in little bags that hold 1414 pound. How many little bags of candy can he fill from the bulk package?

420.

Portions Kristen has 3434 yards of ribbon that she wants to cut into 66 equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be?

Writing Exercises

421.

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning.

422.

Give an example from everyday life that demonstrates how 12·23is13.12·23is13.

423.

Explain how you find the reciprocal of a fraction.

424.

Explain how you find the reciprocal of a negative number.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A table is shown that is made up of four columns and seven rows. The first row reads “I can…” in the first column, “Confidently” in the second column, “With some help” in the third column and “No – I don’t get it” in the last column. The next row down in the first column reads “find equivalent fractions”, under this reads “simplify fractions”, under this reads “multiply fractions”, under this reads “divide fractions”, under this reads “Simplify expressions written with a fraction bar” and under this reads “translate phrases to expressions with fractions.”

After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

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